The quantitative analysis of interactions takes bioinformatics to the next higher dimension: we go from 1D to 2D with graph theory.

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2 The human protein-protein interaction network of aging-associated genes. A total of 261 aging-associated genes were assembled using the GenAge Human Database. Protein-protein interactions of the human interactome were collected from the 8.0 version of the STRING database using physical contacts only. The network was visualized using the Cytoscape program. The degree (number of neighbors) of nodes is represented by the size of the circle and the font. Note the high number of signaling pathway proteins among hubs (nodes with degrees - and therefore size - much greater than average), exemplified by the MAPK/ERK and PI3K/AKT proteins. Figure and caption from: Simkó G, Gyurkó D, Veres DV, Nánási T, and Peter Csermely P (2009) Network strategies to understand the aging process and help age-related drug design. Genome Med. 2009; 1(9): 90 Plots like these have become ubiquitous in the litearture. They have an undeniable aesthetic quality, but the information about biological processes is limited. The authors mention the high number of signalling pathway proteins among the nodes of large degree, but they have not (i) mentioned how a signalling pathway protein is defined, (ii) described in detail how those aging-related proteins were selected in the first place, (iii) compared this fraction against the fraction of signalling pathway proteins among all genes, (iv) quantified their finding, (v) emphasized this class of proteins by colour in the plot. Plotting networks is in general not a good way to analyze them: the inference process has to go the other way around: analyze the network with computational means, then visualize the results in a plot. This plot may or may not be a graph similar to the one above, in this particular case a simple boxplot of node-degree by GO biological process category would have been more effective. Network plots show interactions, but no hypothesis about those interactions has been suggested that the graph could help us visualize. Let us thus discuss the backgrounds of graph theory and useful measures that we can apply towards biological inference from relationship data. 2

3 The quantitative analysis of interactions takes bioinformatics to the next higher dimension: we go from 1D to 2D with graph theory. 3

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9 If we have a directed graph, a node may have incoming edges and outgoing edges. In that case, we distinguish between in-degree and out-degree of the node. We can conceptualize that nodes with a high degree lie in the centre of a network, and that nodes with degree 1 constitute the boundary of the network. Therefore the degree of a node is also a topological measure of the graph, i.e. it describes the contribution of nodes to the overall shape of the graph: we call this interpretation degree centrality. 9

10 Degree distributions give us a way to reason about the circumstances that could have produced a network of interactions in the real world. Barabasi and Albert showed scale-free properties for movie-actor networks, pages in the WWW, and the electric power grid. 10

11 It was soon appreciated, that many biological networks also have scale-free properties. This is not trivial, and begs the question what the WWW and metabolic networks could have in common with power-grid layouts and developmental signalling pathways. 11

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14 Note that the degree-distribution histograms (frequency is on a log-scale) are very characteristic of the generative process. 14

15 The problem here is that it is not obvious why protein-protein interactions should be subject to preferential attachment. Nor is it entirely clear whether actual interaction graphs are indeed scale-free. I do not agree with Barabasi and Albert that the model indicates that growth and preferential attachment play an important role in network development. This would be confusing correlation with causation. They have indeed shown that a model, based on preferential attachment, can reproduce characteristics of many networks. However we must keep an open mind about whether there are other mechanisms that also could give rise to scale-free properties. And indeed there are. Alternative models include: the copy model that creates a scale-free distribution by adding new nodes through copying a fraction of the links of an existing node. Rewiring of random networks towards game-theoretic optimal objective functions also creates scale-free networks. Hierarchical network models are scale-free, as are hyperbolic geometric graphs. All of these have much more straightforward biological analogies than preferential attachment. Besides, the actual networks of biology are not necessarily formed by optimization with a single mechanism towards a common objective function, but are the result of a messy, stochastic, vaguely conserved process of achieving sufficiency of purpose. See also: 15

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19 Besides degree distributions, there are several other approaches to the quantitative analysis of biological networks. 19

20 How many diameters does this graph have? I say: six. 20

21 Dijkstra s algortihm finds a shortest path between two nodes. (i) Suppose we want to find the shortest path between the red ( origin ) and the green ( target ) node. (ii) We set the distance of the origin to 0, all other distances to. The origin is our current node. (iii) Next we collect all neighbors of the current node, and set their distance to one-more than the distance of the current node. (iv) We repeat what we did in (iii) by considering all neighbors in turn. However, we never consider a node that we visited previously, i.e. we include only nodes whose distance is. (v) We repeat what we did in (iii) one more time, This is a loop. Lo-and-behold, this time we encounter the target node. (vi) From the target node, we look for a node whose distance is one-less and add it to the shortest path. We repeat this, always going to a node closer to the origin, and adding that to the path. This is called backtracking. Once we have reached the origin, the shortest path is defined. This is generally considered an O( V 2 ) algorithm, but the implementation based on a min-priority queue implemented by a Fibonacci heap runs in O( E + V \log V ) Fredman ML and Tarjan RE. (1984). Fibonacci heaps and their uses in improved network optimization algorithms. 25th Annual Symposium on Foundations of Computer Science. IEEE. pp

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23 Note that in general, betweeness centrality and degree centrality go in the same direction nodes with high degree tend to have high C b values. But the relationship is not absolute e.g. the second highest C b value, 94, is in a node with degree 3, like two other nodes who only have a C b of 56. This node also has a higher C b than the node with C b 73, which has a degree of 4. Stress centrality is a related concept: the stress centrality of a node w is the number of shortest paths passing through w. 23

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